In the Euclidean space setting the CLR inequality (REF ) can be deduced from the Cwikel estimates (REF ) by using the Birman-Schwinger principle [1]}, [2]}. In its abstract form due to Birman-Solomyak [3]} (see also Proposition REF  and Corollary REF ) the Birman-Schwinger principle implies that if \(H\)  is a (semi-bounded) non-negative selfadjoint operator on Hilbert space and \(V\)  is a non-positive relatively form-compact perturbation such that \((H+1)^{-1/2}V(H+1)^{-1/2}\in {L}_{p,\infty }\) , \(p>0\) , then
\(N(H+V;\lambda ) \le \big \Vert (H+\lambda )^{-\frac{1}{2}}V(H+\lambda )^{-\frac{1}{2}}\big \Vert _{{L}_{p,\infty }}^p\qquad \forall \lambda <0.\) 
